(1) Let (X, d) and (Y, d') be metric spaces, and let f : X — » Y be continuous with f(X) = Y. Show that if (X,d) is complete and d(x,y) < kd'(f(x),f(y)) for some constant k and all x,y € X, then (Y, d') is complete.

(2) Let A be a non-empty compact subset of X. Prove that there exist points a,b € A such that d(a, b) = sup{d(x,y) : x,y € A}.

(1) You need to show that every Cauchy sequence in Y converges. You are told that f is surjective, so a Cauchy sequence in Y must be of the form for some sequence in X. Use the information in the question to deduce that converges to some point , and conclude that converges to f(z).

(2) Let . By the definition of supremum, there are sequences in A such that as . Since A is compact, has a convergent subsequence. Replacing by this subsequence, you can assume that converges, to a say. Then do the same for the other sequence , so that this also converges, to b say. Then conclude that .